Solving Absolute Value Equations and Inequalities

What? A lesson already?

Yes
Any way, you may wonder, what an absolute value is. Well the Absolute Value of a number is the distance from 0 to that number on a number line. ITs written like this: | x |


If c > 0 then | ax+b | = c. If c < 0 then it does not work. You can never have a negative absolute value. 
To solve an absolute value problem you have to do 2 separate equations

For | 2x-5 | = 9

2x-5 = 9 and 2x-5 = -9

then solve

2x = 14  2x = -4

x = 7 and x = -2

Now practice with some problems.

| x+7 | = 24

| x+3 | = 58

| 7x+9 | = 78 

Now can you figure out how to do this?

4| 5x-9| +23 = 76

It's not that complicated! All you have to do is solve for the absolute value. See?:

4| 5x-9| +23 = 76

4| 5x+9 | = 53

| 5x+9 | = 53/4

Oi vey

Now, you should always now that when solving an Absolute Value, you may get what is known as an Extraneus Solution. Take | x-6 | = -3+2x Why don't you solve this one

When you are finally done you should have x = 3 and x = -3. But if you check your work you will find that -3 does not fit! That means it is an Extraneus Solution. Make sure to check your work.

Lastly we have greater thens and less thens (>, <)

There is an easy way to figure this out.

When you get the two answers you either put OR or AND

You put and when the absolute is less thAND

You put or when the absolute is greatOR

See you tomorrow!